Glossary

The number , sometimes called Euler’s constant (also known as Napier’s constant ) is an important mathematical constant.

$e:=\left((infinite-sum1)\right[𝑛\left]\frac{1}{\left(\left(𝑛\right)!\right)}\right)$

$e=2.718281828459045\mathrm{...}$

A sequence of real numbers is -additive if each number in the sequence, after the initial $2\times 𝑠$ terms of the sequence, has exactly $𝑠$ representations as a sum of two previous numbers.

We call a formula $𝐀$ an alphabetic variant of $𝐁$ (or -equal ; write $(alphaeq𝐀)\times 𝐁$), iff $𝐁$ can be obtained from $𝐀$ by systematically renaming bound variables.

We write the -fold composition of the relation $𝑅$ as ${𝑅}^{𝑛}$ and define it by $\left({𝑅}^1\right):=𝑅$ and $\left({𝑅}^{(𝑖+1)}\right):=\left((bsetst𝑆\circ 𝑅)\right[𝑅,𝑋\left]𝑆\in \left({𝑅}^{𝑖}\right)\right)$

An integer whose prime factors have exponents at least $𝑘$ is called a -powerful number , -ful number , or -full number .

An integer whose prime factors have exponents at least $𝑘$ is called a -powerful number , -ful number , or -full number .

A number $𝑛$ is called multiperfect or -perfect for a given natural number $𝑘$, if and only if the sum of all positive divisors of $𝑛$ is equal to $𝑘\times 𝑛$.

An integer whose prime factors have exponents at least $𝑘$ is called a -powerful number , -ful number , or -full number .

A natural number $𝑛$ is called -powersmooth if all prime powers ${\left(𝑝\right)}^{𝑘}$ dividing $𝑛$ satisfy: $\left({\left(𝑝\right)}^{𝑘}\right)<𝐵$

If in a graph $𝐺$ all the vertices have the same degree, say $𝑘$, then $𝐺$ is called -regular , or simply regular .

A -rough number is a positive integer whose prime factors are all greater than or equal to $𝑘$.

A -simplex is a $𝑘$-dimensional convex polytope which is the convex hull of $𝑘+1$ affinely independent points in ${𝐄}^{𝑘}$.

A positive integer is called -smooth if none of its prime factors is greater than $𝐵\in \mathbb{N}$.

The -Ulam numbers $U_{𝑛}$ form an integer sequence . The -Ulam sequence starts with $\left(U_1\right)=𝑢$ and $\left(U_2\right)=𝑣$. Then for $𝑛>2$, $U_{𝑛}$ is defined to be the smallest integer that is the sum of two distinct earlier terms in exactly one way.

The -Ulam numbers $U_{𝑛}$ form an integer sequence . The -Ulam sequence starts with $\left(U_1\right)=𝑢$ and $\left(U_2\right)=𝑣$. Then for $𝑛>2$, $U_{𝑛}$ is defined to be the smallest integer that is the sum of two distinct earlier terms in exactly one way.

A semiprime (also called biprime or 2-almost prime , or pq number ) is a natural number that is the product of two (not necessarily distinct) prime numbers .

A powerful number is the product of a square and a cube. Powerful numbers are also known as squareful numbers , square-full numbers , or 2-full numbers .

A powerful number is a positive integer $𝑚$ such that for every prime number $𝑝$ dividing $𝑚$, ${\left(𝑝\right)}^2$ also divides $𝑚$.

Aban numbers are defined as numbers whose English names do not contain the letter “a” (i.e., “a” is banned). The first numbers are $1,\mathrm{...},999$, $1000000,\mathrm{...},1000999$ and $2000000,\mathrm{...},2000999$.

ABC Conjecture : For every $𝜀>0$, there exist only finitely many triples $⟨𝑎,𝑏,𝑐⟩$ of coprime positive integers with $(𝑎+𝑏)=𝑐$ such that

$(multi-relation-expression\left(q𝑎𝑏𝑐\right)=\frac{(log\left(𝑐\right))}{(log\left((rad\left((𝑎\times 𝑏\times 𝑐)\right))\right))}>1+𝜀)$

ABC Conjecture : For every $𝜀>0$, there exist only finitely many triples $⟨𝑎,𝑏,𝑐⟩$ of coprime natural numbers , with $(𝑎+𝑏)=𝑐$, such that

$𝑐>(rad\left(\left({\left((𝑎\times 𝑏\times 𝑐)\right)}^{(1+𝜀)}\right)\right))$

An group whose operation is commutative is called a commutative group or a Abelian group .

An Abelian group is a group $𝑀$ whose operation is commutative .

An monoid whose operation is commutative is called a commutative monoid or (more rarely) a Abelian monoid .

Given a vector space $𝑉$ over a subfield $𝐹$ of the complex numbers , a norm on $𝑉$ is a function $\parallel \cdot \parallel :𝑉\to ℝ$ such that for all $𝑎\in 𝐹$ and $𝑢,𝑣\in 𝑉$

1. 1.

   $(\parallel (𝑎\cdot 𝑣)\parallel )=(\left(\left|𝑎\right|\right)\times (\parallel 𝑣\parallel ))$ ( absolute homogeneity or absolute scalability ).

2. 2.

   $(\parallel (𝑢+𝑣)\parallel )\le ((\parallel 𝑢\parallel )+(\parallel 𝑣\parallel ))$ ( triangle inequality or subadditivity ).

3. 3.

   If $(\parallel 𝑣\parallel )=0$, then $𝑣$ is the zero vector ( separates points ).

We call the pair $⟨𝑉,\parallel \cdot \parallel ⟩$ a normed vector space with norm $\parallel \cdot \parallel$

Given a vector space $𝑉$ over a subfield $𝐹$ of the complex numbers , a norm on $𝑉$ is a function $\parallel \cdot \parallel :𝑉\to ℝ$ such that for all $𝑎\in 𝐹$ and $𝑢,𝑣\in 𝑉$

1. 1.

   $(\parallel (𝑎\cdot 𝑣)\parallel )=(\left(\left|𝑎\right|\right)\times (\parallel 𝑣\parallel ))$ ( absolute homogeneity or absolute scalability ).

2. 2.

   $(\parallel (𝑢+𝑣)\parallel )\le ((\parallel 𝑢\parallel )+(\parallel 𝑣\parallel ))$ ( triangle inequality or subadditivity ).

3. 3.

   If $(\parallel 𝑣\parallel )=0$, then $𝑣$ is the zero vector ( separates points ).

We call the pair $⟨𝑉,\parallel \cdot \parallel ⟩$ a normed vector space with norm $\parallel \cdot \parallel$

The absolute value $\left|𝑟\right|$ of a real number $𝑟$ is defined as $(defined-piecewise(piece𝑟𝑟\ge 0)\times \left(\left(𝑟\right)sonst\right))$.

A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base.

A number is absolutely non-normal or absolutely abnormal if it is not simply normal in any base.

An abstract reduction system (or abstract rewriting system , or ARS ) $⟨𝐴,𝑅⟩$ consists of a set $𝐴$ together with a relation $𝑅\subseteq \left({𝐴}^2\right)$. The relation $𝑅$ is written as ${\to }_{𝑅}^1$ or simply as ${\to }^1$.

The transitive-reflexive closure of ${\to }^1$ is denoted by ${\to }^{*}$. To distinguish the original relation ${\to }^1$ it is sometimes written as ${\to }^1$.

An abstract reduction system (or abstract rewriting system , or ARS ) $⟨𝐴,𝑅⟩$ consists of a set $𝐴$ together with a relation $𝑅\subseteq \left({𝐴}^2\right)$. The relation $𝑅$ is written as ${\to }_{𝑅}^1$ or simply as ${\to }^1$.

The transitive-reflexive closure of ${\to }^1$ is denoted by ${\to }^{*}$. To distinguish the original relation ${\to }^1$ it is sometimes written as ${\to }^1$.

The abundance of a positive integer $𝑛$ is the value $\left(𝜎1\left(𝑛\right)\right)-(2\times 𝑛)$.

The abundancy of a number $𝑛$ is defined as the ratio $\frac{\left(𝜎1\left(𝑛\right)\right)}{𝑛}$, where $𝜎1\left(𝑛\right)$ is the sum-of-divisors function .

An abundant number or excessive number is a natural number for which the sum of its proper divisors is greater than the number itself.

An Achilles number is a natural number that is powerful but not a perfect power.

Real addition , subtraction , multiplication , division , and exponentiation are extended to the real numbers , so that they respect limits.

Addition is extended to the integers by defining the sum as

$(𝑎+𝑏)=(defined-piecewise(piece\left(\left|𝑎\right|\right)+\left(\left|𝑏\right|\right)𝑎,𝑏\in \mathbb{N})\times (piece-\left((\left(\left|𝑎\right|\right)+\left(\left|𝑏\right|\right))\right)𝑎,𝑏\in {\mathbb{Z}}^{-})\times (piece\left(\left|𝑎\right|\right)+(-(\left(\left|𝑏\right|\right)\left)\right)𝑎\ge 𝑏)\times (piece\left(\left|𝑏\right|\right)+(-(\left(\left|𝑎\right|\right)\left)\right)𝑎<𝑏))$

Addition $+$ computes the sum of $𝑎+𝑏$ of natural numbers $𝑎$ and $𝑏$. It defined by the equations $(𝑥+0)=𝑥$ and $(𝑥+(s\left(𝑦\right)))=\left(s𝑥+𝑦\right)$, where $𝑠$ is the successor function.

Addition on complex numbers , is defined as $((𝑎+𝑏i)+(𝑐+𝑑i))=((𝑎+𝑐)+(\left(𝑏\right)\times 𝑑)i)$.

The number of times the digits must be summed to reach the digital sum is called a number’s additive persistence .

2 Additon $+$ computes the sum of $𝑎+𝑏$ of numbers $𝑎$ and $𝑏$.

Two distinct edges $𝑒$ and $𝑓$ are adjacent if they have an end in common.

Two vertices $𝑥$ and $𝑦$ in a graph $𝐺$ are adjacent , or neighbours , if $⟨𝑥,𝑦⟩$ is an edge of $𝐺$.

Let $𝐾$ be a field , then we call an element $𝑎\in 𝐾$ algebraic , iff $𝑎$ is a root of some non-zero polynomial with coefficients in $𝐾$.

An algebraically closed field $𝐹$ contains a root for every non-constant polynomial in $𝐹\left[𝑋\right]$.

An aliquot sequence starting with a positive integer $𝑘$ is a recursive sequence in which each term is the sum of the proper divisors of the previous term.

The aliquot sum $s\left(𝑛\right)$ is defined as the sum of the proper divisors of $𝑛$.

A natural number is called $𝑘$- almost prime if it has exactly $𝑘$ prime factors , counted with multiplicity.

An alphabet $𝐴$ is a finite set; we call each element $𝑎\in 𝐴$ a character , and an $𝑛$-tuple of $𝑠\in \left({𝐴}^{𝑛}\right)$ a word (or string ) of over $𝐴$. We will often write a string $⟨\left({𝑐}_1,\cdots ,{𝑐}_{𝑛}\right)⟩$ as ${𝑐}_1...{𝑐}_{𝑛}$ or even as ${𝑐}_1...{𝑐}_{𝑛}$. We write the empty word ( empty string ) in ${𝐴}^0$ with $ϵ$.

We call a formula $𝐀$ an alphabetic variant of $𝐁$ (or -equal ; write $(alphaeq𝐀)\times 𝐁$), iff $𝐁$ can be obtained from $𝐀$ by systematically renaming bound variables.

An (language depended) alphamagic square is a magic square in which the number of letters in the name of each number in the square generates another magic square.

The alternating harmonic series is the series

$\left((infinite-sum1)\right[𝑛\left](multi-relation-expression\frac{\left({\left(\left(1\right)\right)}^{(𝑛+1)}\right)}{𝑛}=((((1-\left(\frac{1}{2}\right))+\left(\frac{1}{3}\right))-\left(\frac{1}{4}\right))+\left(\frac{1}{5}\right))-\cdots =log\left(2\right))\right)$

Amicable numbers are two different numbers so related that the sum of the divisors (without the numbers themselves) of each is equal to the other number.

The smallest pair of amicable numbers is $⟨220,284⟩$.

Let . A function $𝑓:𝑈\to ℂ$ is analytic if $𝑓$ has a Taylor series about each point $𝑥\in 𝑈$ that converges pointwise to the function $𝑓$ in an open neighborhood of $𝑥$.

An antimagic square of order $𝑛$ is an arrangement of the numbers $1$ to ${\left(𝑛\right)}^2$ in a square, such that the sums of the $𝑛$ rows, the $𝑛$ columns and the two diagonals form a sequence of $(2\times 𝑛)+2$ consecutive integers .

We call two mathematical objects $𝑎$ and $𝑏$ approximately equal , (written $𝑎\sim 𝑏$), iff they are discerned only by properties of less relevance in the current situation.

A group $𝐺$ with a total order $<$ on its elements is called Archimedean if for all $𝑔\in 𝐺$ there is an $ℎ\in 𝐺$, such that $𝑔<ℎ$.

A graph is a pair $⟨𝑉,𝐸⟩$ such that $𝑉$ is a set and $𝐸\subseteq (𝑉\times 𝑉)$ is a subset of the set of pairs from $𝑉$. We call $𝑉$ the vertices (or nodes , point s, junctions ) and $𝐸$ the edges (or lines , branches , arcs ) of $𝐺$.

In number theory , an arithmetic , arithmetical , or number-theoretic function is a real or complex -valued function $\left(f𝑛\right)$ defined on the set of natural numbers .

The identity function is called $𝐼$: $\left(I\left(𝑛\right)\right)=𝑛$

The following symbols are used:

$\sum _p$	$\prod _p$	the sum or product is over all prime numbers
$\underset{p}{{\sum }^k}$	$\underset{p}{{\prod }^k}$	the sum or product is over all prime powers $𝑘>0$
$\sum _{d|𝑛}$	$\prod _{d|𝑛}$	the sum or product is over all positive divisors of $𝑛$
$\sum _{p|𝑛}$	$\prod _{p|𝑛}$	the sum or product is over all prime divisors of $𝑛$
$\sum _{p^k|𝑛}$	$\prod _{p^k|𝑛}$	the sum or product is over all prime powers dividing $𝑛$

In number theory , an arithmetic , arithmetical , or number-theoretic function is a real or complex -valued function $\left(f𝑛\right)$ defined on the set of natural numbers .

The identity function is called $𝐼$: $\left(I\left(𝑛\right)\right)=𝑛$

The following symbols are used:

$\sum _p$	$\prod _p$	the sum or product is over all prime numbers
$\underset{p}{{\sum }^k}$	$\underset{p}{{\prod }^k}$	the sum or product is over all prime powers $𝑘>0$
$\sum _{d|𝑛}$	$\prod _{d|𝑛}$	the sum or product is over all positive divisors of $𝑛$
$\sum _{p|𝑛}$	$\prod _{p|𝑛}$	the sum or product is over all prime divisors of $𝑛$
$\sum _{p^k|𝑛}$	$\prod _{p^k|𝑛}$	the sum or product is over all prime powers dividing $𝑛$

An abstract reduction system (or abstract rewriting system , or ARS ) $⟨𝐴,𝑅⟩$ consists of a set $𝐴$ together with a relation $𝑅\subseteq \left({𝐴}^2\right)$. The relation $𝑅$ is written as ${\to }_{𝑅}^1$ or simply as ${\to }^1$.

The transitive-reflexive closure of ${\to }^1$ is denoted by ${\to }^{*}$. To distinguish the original relation ${\to }^1$ it is sometimes written as ${\to }^1$.

An aspiring number is a positive integer that are not perfect , but have an eventually repeating aliquot sequence of period $1$.

. We call $\circ$ associative over $G$, iff $((𝑎\circ 𝑏)\circ 𝑐)=(𝑎\circ (𝑏\circ 𝑐))$ for all $𝑎,𝑏,𝑐\in G$.

An associative magic square is a magic square for which every pair of numbers symmetrically opposite to the center sum up to the same value.

A subset $𝐴$ of positive integers has an asymptotic density (or natural density ) $d\left(𝐴\right)$, where $(multi-relation-expression0<d\left(𝐴\right)<1)$, if the limit exists

$(d\left(𝐴\right)):=𝐴\left({lim}_{𝑛\to \infty }\left(\frac{(𝑎\times 𝑛)}{𝑛}\right)\right)$

$𝑎\times 𝑛$ is the number of elements of $𝐴$ less than or equal to $𝑛$.

A atlas for a topological space $𝑀$ is a collection of charts on $𝑀$ such that their domains cover $𝑀$.

The degree (or valency ) $d\left(𝑣\right)$ of a vertex $𝑣$ is the number $n\left((𝐸\times 𝑣)\right)$ of edges at $𝑣$, this equal to the number of neighbours of $𝑣$. A vertex of degree $0$ is called isolated vertex . The number $\left(\delta \left(𝐺\right)\right):=𝐺(min\left(\left((bsetstd\left(𝑣\right))\right[𝑣\left]𝑣\in 𝑉\right)\right))$ is the minimum degree of $𝐺$, the number $\left(\Delta \left(𝐺\right)\right):=𝐺(max\left(\left((bsetstd\left(𝑣\right))\right[𝑣\left]𝑣\in 𝑉\right)\right))$ is the maximum degree of $𝐺$. The number $\left(d\left(𝐺\right)\right):=𝐺\left((errorfragments\frac{1}{(n\left(𝑉\right))},\left((SumInColl𝑉)\right[𝑣\left]d\left(𝑣\right)\right))\right)$ is the average degree of $𝐺$.

The Bailey-Borwein-Plouffe formula ( BBP formula ) is a series for the computation of $\pi$:

$\pi :=\left((infinite-sum0)\right[𝑘\left]\left(\frac{1}{\left({\left(16\right)}^{𝑘}\right)}\right)\times (\left(\frac{4}{((8\times 𝑘)+1)}\right)-\left(\frac{2}{((8\times 𝑘)+4)}\right)-\left(\frac{1}{((8\times 𝑘)+5)}\right)-\left(\frac{1}{((8\times 𝑘)+6)}\right))\right)$

A balanced prime is a prime number that is equal to the arithmetic mean of the nearest primes above and below.

$\left(p_{𝑛}\right)=\left(\frac{(\left(p_{(𝑛-1)}\right)+\left(p_{(𝑛+1)}\right))}{2}\right)$

where $p_{𝑛}$ is the $𝑛$th prime number.

A Banach space is a normed vector space $𝑉$ that is complete in the metric induced by its norm.

A number $𝑛$ is called a barrier of a number-theoretic function $𝑓\times 𝑚$ if, for all $𝑚>𝑛$, $(𝑚+(𝑓\times 𝑚))<𝑛$.

A positional number system is a numeral system that uses positional notation (also called place-value notation ) to encode a number by a sequence of digits .

A positional number system for a number system $𝑁$ is given as a pair $𝑃:=(⟨𝐷,𝜑⟩)$, where $𝐷$ is a finite set $𝐷$ of digit (s we call $𝑏:=(n\left(𝐷\right))$ the base or radix of $𝑃$) and a injective mapping from $𝐷$ to $𝑁$.

positional notation extends $𝜑$ to a bijective mapping from finite sequences over $𝐷$ to $𝑁$ using the arithmetics of $𝑁$ by interpreting a finite sequence ${𝑎}_0,\mathrm{...},{𝑎}_{𝑛}$ as a sum of successive powers ${\left(𝑏\right)}^{𝑖}$ multiplied by $\left(\phi {𝑎}_{𝑖}\right)$. Details vary with $𝑁$.